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parsiad
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Here's an approach that's easy to code (but FAR from the fastest). Let $ f(T,S,v,K) $ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).

Here's an approach that's easy to code. Let $ f(T,S,v,K) $ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).

Here's an approach that's easy to code (but FAR from the fastest). Let $ f(T,S,v,K) $ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).

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parsiad
parsiad
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Do you mean determiningHere's an approach that's easy to code. Let $$ f(S_{0}):=\mathbb{E}\left[\sup_{\tau\in\mathcal{T}_{tT}}e^{-r\tau}\max\left\{ S_{\tau}-K,0\right\} \right] $$$ f(T,S,v,K) $ where $\mathcal{T}_{tT}$ isdenote the setprice of all stopping times taking values in $\left[t,T\right]$ a.s.? ($t>0$ for a forward start) If this is what you mean, I believe any of European call in the standard approaches wouldHeston model with time-to-expire work here: you can price using a a PDE approach$T$, Monte Carlo and linear least squaresinitial price $S$, characteristic function and FFTinitial volatility $v$, etc.strike $K$. All you haveFirst, to do is make sure thatuse the American constraint is removed at alltower property to transform the pricing problem: times\begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above $<t$(the advantage here is that you can use existing procedures to compute $f$).

Do you mean determining $$ f(S_{0}):=\mathbb{E}\left[\sup_{\tau\in\mathcal{T}_{tT}}e^{-r\tau}\max\left\{ S_{\tau}-K,0\right\} \right] $$ where $\mathcal{T}_{tT}$ is the set of all stopping times taking values in $\left[t,T\right]$ a.s.? ($t>0$ for a forward start) If this is what you mean, I believe any of the standard approaches would work here: you can price using a a PDE approach, Monte Carlo and linear least squares, characteristic function and FFT, etc.. All you have to do is make sure that the American constraint is removed at all times $<t$.

Here's an approach that's easy to code. Let $ f(T,S,v,K) $ denote the price of a European call in the Heston model with time-to-expire $T$, initial price $S$, initial volatility $v$, strike $K$. First, use the tower property to transform the pricing problem: \begin{align*} V_{0} & =\mathbb{E}\left[e^{-r\left(t+\tau\right)}\left(S_{t+\tau}/S_{t}-K\right)^{+}\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}\mathbb{E}\left[e^{-r\tau}\left(S_{t+\tau}-KS_{t}\right)^{+}\mid\mathcal{F}_{t}\right]\right]\\ & =\mathbb{E}\left[\frac{e^{-rt}}{S_{t}}f(\tau,S_{t},v_{t},KS_{t})\right]. \end{align*} Now perform a Monte-Carlo simulation to approximate the above (the advantage here is that you can use existing procedures to compute $f$).

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parsiad
parsiad
  • 206
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Do you mean determining $$ f(S_{0}):=\mathbb{E}\left[\sup_{\tau\in\mathcal{T}_{tT}}e^{-r\tau}\max\left\{ S_{\tau}-K,0\right\} \right] $$ where $\mathcal{T}_{tT}$ is the set of all stopping times taking values in $\left[t,T\right]$ a.s.? ($t>0$ for a forward start) If this is what you mean, I believe any of the standard approaches would work here: you can price using a a PDE approach, Monte Carlo and linear least squares, characteristic function and FFT, etc.. All you have to do is make sure that the American constraint is removed at all times $<t$.